Warm UpWarm UpSolve the equation.Solve the equation.

1. 2 5

2. 4 2 10

3. 3 6 8

4. 2 9 3 7

5. 6 3 2 1 15

x

x

x

x

x

answersanswers1.1. x = 7 and x = -3x = 7 and x = -32.2. x = 3 and x = -2x = 3 and x = -23.3. x = 5 and x = 1x = 5 and x = 14.4. x = -2 ½ and x = -6 ½ x = -2 ½ and x = -6 ½ 5.5. x = 1 and x = -2x = 1 and x = -2

3.5 3.5 Solving and Graphing Solving and Graphing

InequalitiesInequalities

► What is an What is an inequality?inequality?

► What are the What are the inequality inequality signs?signs?

► An inequality is a statement An inequality is a statement that shows how two that shows how two expressions compare, using expressions compare, using greater than or less than greater than or less than symbols.symbols.

toequalor than less toequalor an greater th

thanlessangreater th

ExplorationExploration► Write each pair of numbers, Write each pair of numbers,

write an inequality symbol write an inequality symbol to compare them, and to compare them, and perform the indicated steps.perform the indicated steps.

► Add four to each side. Add four to each side. Compare the answers.Compare the answers.

► Subtract five from each Subtract five from each side. Compare the answers.side. Compare the answers.

55 885 < 85 < 8

9 < 129 < 12

4 < 74 < 7

ExplorationExploration► Write each pair of numbers, Write each pair of numbers,

write an inequality symbol write an inequality symbol to compare them, and to compare them, and perform the indicated steps.perform the indicated steps.

► Add ten to each side. Add ten to each side. Compare the answers.Compare the answers.

► Subtract seven from each Subtract seven from each side. Compare the answers.side. Compare the answers.

-3-3 -8-8-3 > -8-3 > -8

7 > 27 > 2

0 > -50 > -5

What did you notice?What did you notice?►Summarize what you noticed about Summarize what you noticed about

the inequality sign after performing the inequality sign after performing the indicated operations.the indicated operations.

Adding or subtracting the same number to Adding or subtracting the same number to both sides of the inequality does not both sides of the inequality does not change the inequality sign.change the inequality sign.

ExplorationExploration► Write each pair of numbers, Write each pair of numbers,

write an inequality symbol write an inequality symbol to compare them, and to compare them, and perform the indicated steps.perform the indicated steps.

► Multiply by four on each Multiply by four on each side. Compare the answers.side. Compare the answers.

► Divide by two on each side. Divide by two on each side. Compare the answers.Compare the answers.

44 -12-124 > -124 > -12

16 > -4816 > -48

8 > -248 > -24

ExplorationExploration► Write each pair of numbers, Write each pair of numbers,

write an inequality symbol write an inequality symbol to compare them, and to compare them, and perform the indicated perform the indicated steps.steps.

► Multiply by negative two on Multiply by negative two on each side. Compare the each side. Compare the answers.answers.

► Divide by negative four on Divide by negative four on each side. Compare the each side. Compare the answers.answers.

-6-6 1010-6 < 10-6 < 10

12 > -2012 > -20

-3 < 5-3 < 5

What did you notice?What did you notice?►Summarize what you noticed about the Summarize what you noticed about the

inequality sign after performing the inequality sign after performing the indicated operation.indicated operation.

Multiplying or dividing by a positive number Multiplying or dividing by a positive number on both sides does not affect the inequality on both sides does not affect the inequality sign.sign.

Multiplying or dividing by a negative number Multiplying or dividing by a negative number on both sides changes the direction of the on both sides changes the direction of the inequality sign.inequality sign.

► Steps for Steps for Solving an Solving an InequalityInequality

1)1) DistributeDistribute2)2) Combine like termsCombine like terms3)3) Move the variable to Move the variable to

one sideone side4)4) Undo addition or Undo addition or

subtractionsubtraction5)5) Undo multiplication or Undo multiplication or

division (if mult. or div. division (if mult. or div. by a neg. change sign)by a neg. change sign)

► Steps for Steps for Graphing an Graphing an InequalityInequality

1)1) Draw a number line Draw a number line including the number in including the number in the solution.the solution.

2)2) Draw a circle around the Draw a circle around the value in the solution.value in the solution.

3)3) Fill in the circle if the Fill in the circle if the value is a part of the value is a part of the solution, leave it open if solution, leave it open if it is not part of the it is not part of the solution.solution.

4)4) Draw an arrow in the Draw an arrow in the direction of the values in direction of the values in the solution.the solution.

ExamplesExamplesSolve and graph each inequality.Solve and graph each inequality.1)1) x + 7 > 13x + 7 > 13

2)2) -21 > x - 18 -21 > x - 18

x > 6 4 5 6 7 8 9

-3 > xx < -3

-5 -4 -3 -2 -1 0

ExamplesExamples► Solve and graph Solve and graph

each inequality.each inequality.3)3)

4)4)

43

50 5

n

x

n -12 -14 -12 -10

-14 -12 -101010x

x

ExamplesExamplesSolve and GraphSolve and Graph5)5) 2x + 1 > 13 2x + 1 > 13

- 1 -1 2x > 12

2 2X > 6

6) 7 – 2t > 21-7 - 7-2t > 14-2 -2t < -7

0 642 8-1-5 0-3-7-9

Try These: solve and graphTry These: solve and graph

7. 3 – 3x + 2 > 5 – 2x7. 3 – 3x + 2 > 5 – 2x8. 4m – 4 8. 4m – 4 << 2 + 4m 2 + 4m9. -3(x – 2) < 3 – (2x + 4)9. -3(x – 2) < 3 – (2x + 4)

Answers7. 0 > x 08. I.M.S9. 7 < x 7

Summary/ReflectionSummary/Reflection►How is solving an inequality alike and How is solving an inequality alike and

different from solving an equation?different from solving an equation?►How will you remember which How will you remember which

direction to draw the arrow on the direction to draw the arrow on the graph?graph?

►How will you remember whether to How will you remember whether to use an open or closed circle?use an open or closed circle?

HomeworkHomework►Worksheet 3.5 Worksheet 3.5